and I'm thinking [tex]r=\frac{F}{cos\theta}[/tex] but I'm not too sure about it.

that can't be right, because F is a force vector (dealing with acceleration) and r is a position vector (which can be treated as a scalar here).

For uniform acceleration to occur, [tex]\vec{\tau}_{net}=\vec{F}_{net}\times\vec{r}=0[/tex], and we already know that [tex]r\neq0[/tex], so [tex]F_{net}[/tex] must equal to 0. Then resolve it into x and y components.

that can't be right, because F is a force vector (dealing with acceleration) and r is a position vector (which can be treated as a scalar here).

For uniform acceleration to occur, [tex]\vec{\tau}_{net}=\vec{F}_{net}\times\vec{r}=0[/tex], and we already know that [tex]r\neq0[/tex], so [tex]F_{net}[/tex] must equal to 0. Then resolve it into x and y components.

Sorry where did you get that equation from? I don't understand what
[tex]\vec{\tau}_{net}[/tex] is.

I'm pretty sure that [tex]\tau[/tex]net is the net torque on the particle P. Since the particle P does undergo uniform circular motion there is a torque on particle P, and he was simply using [tex]\tau[/tex]net = Fnet x r = 0 to show you that since (r) isn't 0, Fnet must be equal to 0 in order to fulfill this equation.

Other posters note, the course Mentallic is doing does not cover cross/dot products, vectors in the "advanced" sense, torque etc . It only allows for elementary calculus to solve the problems.

Mentallic - The hard part is resolving the forces! For each vector, draw a right angled triangle so that one side is purely horizontal and the other is vertical. This need not be done for the always vertical mg force downwards. Find the lengths of the sides using trig and add up vertical components and horizontal components, taking care to have a negative sign when they are in opposite directions.

Then take gabbagabbahey's suggestion - Now that we have an expression for the vertical and horizontal components, what can we equate that to? In circular motion, what is the net force, what direction is it in. Hence, what must the components be equal to? Once you form those equations, the rest is relatively easy.